Definition 4.5 ([1]}, [2]}) 
Let \(\mathcal {G}=(\mathcal {V},\mathcal {E})\)  be a directed graph with \(|\mathcal {V}|=N\) . Let \(\mathcal {X}\)  be an ergodic random walk on \(\mathcal {G}\)  with transition matrix \(\mathbf {P}\)  and unique stationary distribution \(\pi \) . The normalized Laplacian on \(\mathcal {G}\)  is defined by
\(\mathcal {L}=\mathbf {I}-\frac{\Pi ^{1/2}\mathbf {P}\Pi ^{-1/2}+\Pi ^{-1/2}\mathbf {P}^{\top }\Pi ^{1/2}}{2},\) 
